**3. Similarity Transformations**

Introducing the subsequent similarity variables satisfying the continuity equation, for instance:

$$\begin{cases} \begin{array}{l} \upsilon\_{r} = r \frac{\partial F}{\partial z} = \frac{\partial r}{2} \frac{D^{2}}{\Gamma} f'(\lambda), \upsilon\_{\theta} = G(z, t) r = r \Omega\_{1} \frac{D^{2}}{\Gamma} g(\lambda), \\ \upsilon\_{z} = -2F(z, t) = -\frac{\beta D^{2} f(\lambda)}{\Gamma(t)}, \end{array} \\\ \begin{array}{l} B\_{r} = r \frac{\partial M}{\partial z} = \frac{\partial r M\_{0}}{\partial z} m'(\lambda), B\_{\theta} = r N(z, t) = r N\_{0} \frac{D^{2}}{\Gamma} n(\lambda), \\ B\_{\overline{z}} = -2M(z, t) = -\frac{\beta DM\_{0} m(\lambda)}{\Gamma(t)}, \end{array} \\\ \begin{array}{l} \phi(\lambda) = \frac{\breve{\overline{C}} - \breve{\overline{C}}\_{\mu}}{\breve{\overline{C}}\_{l} - \breve{\overline{C}}\_{\mu}}, \chi(\lambda) = \frac{\breve{\overline{C}} - \breve{\overline{\overline{\mu}}}}{\widetilde{\overline{\Gamma}}\_{l} - \breve{\overline{\overline{\mu}}\_{\mu}}}, \lambda = \frac{\overline{z}}{\overline{\Gamma}(t)}. \end{array} \end{cases} \tag{27}$$

where similarity variable is *λ* and *f*(*λ*), *g*(*λ*), *<sup>m</sup>*(*λ*), *<sup>n</sup>*(*λ*), *θ* (*λ*), *φ*(*λ*) and *χ*(*λ*) are nondimensional velocity in axial and tangential direction, the magnetic field in axial and tangential direction, temperature, concentration, and motile density function, respectively. Nowsubstitutingtheabove-mentionedsimilaritytransformationinEquations(6)–(16),

 following coupled, nonlinear ODE's with independent variable (*λ*) obtained as,

$$\begin{split} f^{(iv)}(\eta) = 4R\_Q \left[ 3f^{\prime\prime} - 2\left(\frac{R\_\Omega}{S\_Q}\right)^2 \text{g} \mathbf{g}^{\prime} + 2F\_T^2 (mm^{\prime\prime} + m^\prime m^\prime) - (2f - \lambda)f^{\prime\prime} + 2F\_A^2 \left(\frac{R\_\Omega}{S\_Q}\right)^2 m^\prime \right] \\ - 4K \left[ \frac{2R\_\Omega}{R\_Q} \mathbf{g}^{\prime} \mathbf{g}^{\prime\prime} + \frac{R\_Q}{R\_\Omega} \left[ 2f^{\prime\prime} f^{\prime\prime\prime} + 2\left(f^{\prime\prime} f^{\prime\prime\prime} + f^{\prime} f^{\prime\prime\prime} \right) \right] \right] \end{split} \tag{28}$$

$$\log''(\eta) = 2S\_Q \left[ 2\mathbf{g} + \lambda \mathbf{g}' + 2\mathbf{g}f' - f\mathbf{g}' + 2F\_A F\_T \left( m\mathbf{n}' + m\mathbf{n}' \right) \right] - 2K \left[ \mathbf{g}'(\eta)f''(\eta) - f'(\eta)g''(\eta) \right],\tag{29}$$

$$m'' = \operatorname{Re}\_M \left[ m + \lambda m' + 2mf' - 2fm' \right],\tag{30}$$

$$n'' = \text{Re}\_M \left[ 2n - fn' + \lambda n' + 2\left(\frac{F\_A}{F\_T}\right) mg' \right],\tag{31}$$

$$\left(1+\frac{4}{3}R\_d(1+\left(T\_I-1\right)\overset{\smile}{\theta})^3\right)\overset{\smile}{\theta''}+4Rd(T\_I-1)(1+\left(T\_I-1\right)\overset{\smile}{\theta})^2\overset{\smile}{\theta}^2+\mathbb{S}\_Q\\ \overset{\smile}{\theta}f\overset{\smile}{\theta}^2+\mathbb{T}\_I\overset{\smile}{\theta'}^2+\mathbb{T}\_b\overset{\smile}{\theta'}\overset{\smile}{\phi'}=0,\tag{32}$$

$$\Phi'' + \frac{T\_t}{T\_b} \overset{\smile}{\theta''} + S\_Q S\_M f \phi' - S\_M \sigma (1 + \tilde{\delta} \stackrel{\smile}{\theta}) \frac{n}{\exp \left( - \frac{E}{1 + \tilde{\delta} \stackrel{\smile}{\theta}} \right)} \Phi = 0,\tag{33}$$

$$
\chi'' - S\_Q B\_5 \left(\frac{\lambda}{2}\right) \chi' + B\_5 S\_Q f \chi' - P\_l \left[\chi' \phi' + (\chi + \Phi)\phi''\right] = 0. \tag{34}
$$

where *SQ* represents the squeezed Reynolds number, *R*Ω the rotational Reynolds number, *FA*, *FT*, denote the strength of the magnetic field in axial and azimuthal direction, Re*M* the magnetic Reynolds number, *K* the material parameter of Reiner-Rivlin fluid, *Tb* the Brownian motion, *Pt* the Prandtl number, *Tt* the Thermophoresis parameter, *E* the non-dimensional form of Arrhenius activation energy, *SM* the Schmidt number, *Bs* the bioconvection Schmidt number, *σ* the rate of chemical reaction, *Pl* the Peclet number, *δ* + represents the temperature ratio, *Tr* the temperature ratio parameter, *Rd* the radiation parameter, and Φ the constant number, respectively. They can be written as

$$\begin{cases} \begin{aligned} S\_{Q} &= \frac{\beta D^{2}}{2v}, R\_{\Omega} = \frac{\Omega\_{1} D^{2}}{v}, F\_{T} = \frac{M\_{0}}{D\sqrt{\mu\_{2}\rho}}, F\_{A} = \frac{N\_{0}}{\Omega\_{1}\sqrt{\mu\_{2}\rho}}, K = \frac{\mu\_{e}\Omega}{\mu}, \\\ T\_{b} &= \frac{\tau D\_{b}\left(\breve{\breve{\zeta}\_{l} - \breve{\zeta}\_{u}}{\breve{a}}\right)}{\breve{a}}, T\_{l} = \frac{\tau D\breve{\tau}\left(\breve{\breve{\boldsymbol{\tau}}\_{l} - \breve{\boldsymbol{\tau}}\_{u}\right)}{\breve{a}\breve{\boldsymbol{\tau}}\_{u}}, P\_{l} = \frac{\mu}{\breve{a}}, \breve{\boldsymbol{\alpha}} = \frac{k}{\left(\mu\breve{\boldsymbol{\varepsilon}}\right)\_{p}} S\_{M} = \frac{v}{\mathcal{D}\_{b}}, \\\ B\_{s} = \frac{v}{\mathcal{D}\_{u}}, P\_{l} = \frac{bW\_{\mathrm{mo}}}{D\_{\mathrm{mo}}}, \Phi = \frac{n\_{u}}{\eta\_{l} - u\_{u}}, Bt = \delta\mu\_{2}\nu, \mathrm{Re}\_{M} = R\_{\mathrm{Q}}Bt, R\_{d} = \frac{4\overline{T}\_{u}\mu\_{r}}{\beta\_{r}k}, \\\ E = \frac{E\_{a}}{\overline{\kappa\_{a}T}}, \sigma = \frac{k^{2}\overline{\Gamma}(t)^{2}}{v}, \widetilde{\delta} = \frac{\overline{T}\_{l} - \overline{T}\_{u}}{\overline{\overline{\gamma}\_{u}}}, \tau = \frac{(\rho c)\_{p}}{(\rho c)\_{f}}, T\_{r} = \frac{T\_{l}}{T\_{u}} \end{aligned} \tag{35}$$

where *Bt* represents Batchelor number.

The boundary conditions said in Equations (25) and (26) reduced as

$$\begin{cases} \quad f'(0) = 0, \, f(0) = 0, m(0) = 0, \, \underline{\chi}(0) = 1, \, n(0) = 1, \, \stackrel{\leftarrow}{\theta}(0) = 1, \, \underline{\chi}(0) = 1, \, \underline{\phi}(0) = 1, \\\quad f(1) = \frac{1}{2}, \, m(1) = 1, \, \underline{\chi}(1) = \frac{1}{2}, \, m(1) = 1, \, \underline{\theta}(1) = 0, \, \underline{\phi}(1) = 0, \, \underline{\chi}(1) = 0 \end{cases} \tag{36}$$

where *f* , *g*, *n*, *m*, *θ*, *φ*, *χ* denotes axial velocity and tangential velocity, magnetic field components in the tangential and axial direction, temperature distribution, nanoparticles concentration, motile gyrotactic microorganism profile, .*ξ*(= <sup>Ω</sup>2/Ω1) represents the angular velocity, and its range is in between the rotating plates −1 ≤ . *ξ* ≤ 1. It is beneficial to investigate various revolving flow attributes of rotating plates in the reverse or same direction.

On the upper (moving) plate, the dimensionless torque can be calculated as

$$\hat{T}\_{\rm up} = 2\pi\rho \int\_0^b \left(\frac{\partial v}{\partial z}\right)\_{z=\stackrel{\frown}{\Gamma}(t)} \,\mathrm{d}r,\tag{37}$$

where the plate radius is signified by *b*.

> Using Equation (27) in Equation (37), it becomes

$$
\hat{T}\_{\rm up} = \frac{d\mathcal{g}(1)}{d\lambda},
\tag{38}
$$

where the upper plate torque is designated by *T*ˆup, and the tangential velocity gradient on the upper (moving) plate is *dg*(1)/*dλ*.

In the same fashion, the lower plate torque in dimensionless form is achieved by similar calculation and it becomes for *λ* = 0 as

$$
\hat{\mathcal{I}}\_{\rm lp} = \frac{d\mathbf{g}(0)}{d\lambda}.\tag{39}
$$

### **4. Solution of the Problem by DTM-Padé**

DTM was first introduced by Zhou [55] in an engineering analysis for electric circuit theory for linear and nonlinear problems. It is an extremely powerful method for finding the solutions of magnetohydrodynamics and complex material flow problem. The Differential Transform Method (DTM) is distinct from the conventional higher-order Taylor series scheme. It was also used in combination with Padé approximants very successfully. The purpose of applying Padé-approximation is to improve the convergence rate of series solutions. The reason behind this is that sometimes the DTM fails to converge. That is why most of the researchers' merge DTM and Padé approximation to deal with the high order nonlinear differential equations. The Padé approximation is a rational function that can be thought of as a generalization of a Taylor polynomial. A rational function is the ratio of polynomials. Because these functions only use the elementary arithmetic operations, they are very easy to evaluate numerically. The polynomial in the denominator allows one to approximate functions that have rational singularities All the codes are developed on Mathematica software. The dimensionless Equations (28)–(36) are attained with the help of similar transformations stated in Equation (27), which are solved by virtue of the Differential Transform Method. To proceed further with the DTM technique, let us define *qth* derivative as:

$$F(\lambda) = \frac{1}{q!} \left[ \frac{d^q f}{d\lambda^q} \right]\_{\lambda = \lambda\_0} \, \text{\AA} \tag{40}$$

where *f*(*λ*) are original and *F*(*λ*) represent transformed functions. Now the differential inverse transform *F*(*λ*) can be defined as

$$f(\lambda) = \sum\_{q=0}^{\infty} F(\lambda)(\lambda - \lambda\_0)^q \, \tag{41}$$

The objective of differential transformation has been achieved by the Taylor extension series, and in terms of the finite series, the function *f*(*λ*) can be defined as

$$f(\lambda) \cong \sum\_{q=0}^{k} F(\lambda)(\lambda - \lambda\_0)^q \, \prime \tag{42}$$

The rate of convergence depends upon the value of *k*. Each BVP can be converted to IVP with the replacement of unknown initial conditions. Taking differential transformation of the separate term by term of Equations (28)–(36), the following transformations are attained:

$$\begin{split} f^{\boldsymbol{\nu}} &\rightarrow (1+\lambda)(2+\lambda)f(\lambda+2), \\ f^{\boldsymbol{\nu}^{3}} &\rightarrow \left[ \sum\_{\boldsymbol{\nu}=0}^{\lambda} \binom{\boldsymbol{\omega}}{\boldsymbol{\nu}=0} (\omega+1)(\omega+2)(-\omega+\overline{\boldsymbol{\nu}}+1)(-\widetilde{\boldsymbol{\nu}}+\lambda+1)(-\widetilde{\boldsymbol{\nu}}+\lambda+2)} \right), \\ f^{\boldsymbol{\nu}}f^{\boldsymbol{\nu}} &\rightarrow \left[ \sum\_{\widetilde{\boldsymbol{\nu}}=0}^{\lambda} \binom{\widetilde{\boldsymbol{\nu}}-\widetilde{\boldsymbol{\nu}}+\lambda+2)f(2+\omega)f(-\omega+2+\lambda)}{\sum\_{\widetilde{\boldsymbol{\nu}}=0}^{\lambda} (\omega+1)(1+\omega)(2+\omega)(-\omega+\lambda-\widetilde{\boldsymbol{\nu}}+1)(-\omega+2+\lambda-\widetilde{\boldsymbol{\nu}})} \right) \bigg], \\ f^{\boldsymbol{\nu}}f^{\boldsymbol{\nu}2} &\rightarrow \left[ \sum\_{\widetilde{\boldsymbol{\nu}}=0}^{\lambda} \binom{\lambda-\omega}{(-\widetilde{\boldsymbol{\nu}}+\lambda-\omega+3)f(1+\omega)f(2+\widetilde{\boldsymbol{\nu}})f(-\omega+\lambda-\widetilde{\boldsymbol{\nu}}+3)} \right], \\ f^{\boldsymbol{\nu}}f^{\boldsymbol{\nu}2} &\rightarrow \left[ \sum\_{\widetilde{\boldsymbol{\nu}}=0}^{\lambda} \binom{\widetilde{\boldsymbol{\nu}}+\omega}{(-\widetilde{\boldsymbol{\nu}}+\lambda+2-\omega)(-\widetilde{\boldsymbol{\nu}}+\lambda-\omega+3)f(3+\omega)f(2+\widetilde{\boldsymbol{\nu}})f(-\widetilde{\boldsymbol{\nu}}+\lambda-\omega+3)} \right), \end{split} \tag{43}$$

*g* → *g*(*l*), *<sup>λ</sup>g* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*ε*(*ω*)*g*(− *ω* + 1 + <sup>ň</sup>)), ⎫ ⎬ ⎭ (44) *f g* → ň ∑ *<sup>ω</sup>*=0 (− *ω* + 1 + <sup>ň</sup>)*f*(*ω*)*g*(− *ω* + 1 + <sup>ň</sup>), *g f* → ň ∑ *<sup>ω</sup>*=0 (− *ω* + 1 + <sup>ň</sup>)*g*(*ω*)*f*(− *ω* + 1 + <sup>ň</sup>), *g f g* → ň ∑ *<sup>ω</sup>*=0 (*ω* + 1)( − *ω* + 1 + ň)( − *ω* + 2 + ň)*f*(1 + *ω*)*g*(<sup>1</sup> + *<sup>ω</sup>*)*g*(− *ω* + 2 + <sup>ň</sup>), *g g f* → ň ∑ *<sup>ω</sup>*=0 (1 + *ω*)( *ω* + 2)( − *ω* + 1 + ň)( − *ω* + 2 + ň)( − *ω* + ň + <sup>3</sup>)*g*(<sup>1</sup> + *ω*)*g*(<sup>2</sup> + *ω*) *g*(− *ω* + ň + <sup>3</sup>), *g f f* → ň ∑ *<sup>m</sup>*=0 (1 + *ω*)( − *ω* + ň + 1)( − *ω* + 2 + ň)*f*(1 + *ω*)*g*(<sup>1</sup> + *<sup>ω</sup>*)*f*(− *ω* + 2 + <sup>ň</sup>), *f g f* → ň ∑ *<sup>ω</sup>*=0 (1 + *ω*)(<sup>2</sup> + *ω*)( − *ω* + 1 + ň)( − *ω* + 2 + ň)( − *ω* + ň + <sup>3</sup>)*g*(*ω* + 1)*f*(2 + *ω*) *f*(− *ω* + 3 + <sup>ň</sup>), *f g*2 → ň ∑ *<sup>ω</sup>*=0 ⎛ ⎝ ň ∑ *<sup>υ</sup>*<sup>+</sup>=0 (1 + *ω*)(<sup>2</sup> + *ω*)(<sup>1</sup> − *ω* + *υ*<sup>+</sup>)( −*υ*<sup>+</sup>+ 1 + <sup>ň</sup>)*g*(−*υ*<sup>+</sup><sup>+</sup> 1 + ň)*f*(2 + *ω*) *g*(− *ω* + 1 + ň) ⎞ ⎠, *g* 2 *f* → ň ∑ *<sup>ω</sup>*=0 ⎛ ⎝ ň−*<sup>ω</sup>* ∑ *<sup>υ</sup>*<sup>+</sup>=0 (*ω* + 1)(2 + *ω*)(*υ*<sup>+</sup><sup>+</sup> <sup>1</sup>)(*υ*<sup>+</sup><sup>+</sup> 2)( −*υ*<sup>+</sup>+ 1 − *ω* + ň)( − *ω* + 2 + ň − *υ*+) *g*(<sup>2</sup> + *ω*)*f*(*q* + <sup>2</sup>)*g*(− *ω* + 2 + ň − *υ*+) ⎞ ⎠, *f* 2 *g* → ň ∑ *<sup>υ</sup>*<sup>+</sup>=0 ⎛ ⎝ *υ*+ ∑ *<sup>ω</sup>*=0 (1 + *ω*)(<sup>2</sup> + *ω*)( − *ω* + 1 + *υ*<sup>+</sup>)( −*υ*<sup>+</sup>+ 1 + <sup>ň</sup>)*f*(−*υ*<sup>+</sup><sup>+</sup> 1 + <sup>ň</sup>)*g*(<sup>2</sup> + *ω*) *f*(− *ω* + ň + 1) ⎞ ⎠, ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (45) *m m* → ň ∑ *<sup>ω</sup>*=0 (*ω* + 1)(2 + *ω*)( − *ω* + 1 + ň)( − *ω* + 2 + <sup>ň</sup>)*m*(*ω* + <sup>1</sup>)*m*(− *ω* + 2 + <sup>ň</sup>), *λ m* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*ε*(*ω*)*m*(− *ω* + ň + <sup>1</sup>)), *m f* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*m*(*ω*)*f*(− *ω* + 1 + <sup>ň</sup>)), *f m* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*f*(*ω*)*m*(− *ω* + 1 + <sup>ň</sup>)), *mg* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*m*(*ω*)*g*(− *ω* + 1 + <sup>ň</sup>)), ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (46) *nn* → ň ∑ *<sup>m</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*n*(*ω*)*n*(− *ω* + 1 + <sup>ň</sup>)), *f n* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*f*(*ω*)*n*(− *ω* + 1 + <sup>ň</sup>)), *λn* → ň ∑ *<sup>ω</sup>*=0 (( − *ω* + 1 + <sup>ň</sup>)*ε*(*ω*)*n*(− *ω* + 1 + <sup>ň</sup>)), ⎫ ⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭ (47) *f θ* → ň ∑ *<sup>ω</sup>*=0 (− *ω* + 1 + <sup>ň</sup>)*f*(*ω*) *θ* (− *ω* + 1 + ň) , *θ* 2 → ň ∑ *<sup>ω</sup>*=0 (1 + *ω*)( − *ω* + 1 + ň) *θ* (1 + *ω*) *θ* (1 − *ω* + ň) , ⎫ ⎪⎪⎬ ⎪⎪⎭ (48) *θ φ* → ň ∑ *<sup>ω</sup>*=0 (1 + *ω*)( − *ω* + 1 + ň) *θ* (1 + *<sup>ω</sup>*)*φ*(− *ω* + 1 + ň) , *f φ* → ň ∑ *<sup>ω</sup>*=0(( − *ω* + 1 + <sup>ň</sup>)*f*(*ω*)*φ*(− *ω* + 1 + <sup>ň</sup>)), ⎫ ⎪⎪⎬ ⎪⎪⎭ (49)

$$\begin{array}{c} \lambda \chi' \to \mathop{\Sigma}\_{\omega=0} \left( ( - \omega + 1 + \lambda ) \varepsilon(\omega) \chi(-\omega + 1 + \lambda) \right), \\ f \chi' \to \mathop{\Sigma}\_{\omega=0} \left( ( - \omega + 1 + \lambda ) f(\omega) \chi(-\omega + 1 + \lambda) \right), \\ \chi' \phi' \to \mathop{\Sigma}\_{\omega=0} \left( ( 1 + \omega )( - \omega + 1 + \lambda ) \chi(\omega + 1) \phi(-\omega + 1 + \lambda) \right), \\ \chi \phi'' \to \mathop{\Sigma}\_{\omega=0} \left( ( - \omega + 1 + \lambda )( - \omega + 2 + \lambda) \chi(\omega) \phi(-\omega + 2 + \lambda) \right), \end{array} \tag{50}$$

where *f*(*l*), *g*(*l*), *<sup>m</sup>*(*l*), *<sup>n</sup>*(*l*), *θ* (*l*), *φ*(*l*) and *χ*(*l*) are the transformed function of *f*(*λ*), *g*(*λ*), *<sup>m</sup>*(*λ*), *<sup>n</sup>*(*λ*), *<sup>θ</sup>*(*λ*), *φ*(*λ*) and *<sup>χ</sup>*(*λ*), respectively, and are expressed as

$$f(\lambda) = \sum\_{l=0}^{\infty} f(l)\lambda^l,\tag{51}$$

$$\mathbf{g}(\boldsymbol{\lambda}) = \sum\_{l=0}^{\infty} \mathbf{g}(l) \boldsymbol{\lambda}^{l} \,. \tag{52}$$

$$m(\lambda) = \sum\_{l=0}^{\infty} m(l)\lambda^{l},\tag{53}$$

$$m(\lambda) = \sum\_{l=0}^{\infty} n(l)\lambda^l \,\_{\prime} \tag{54}$$

$$\stackrel{\smile}{\theta}(\lambda) = \sum\_{l=0}^{\infty} \stackrel{\smile}{\theta}(l) \lambda^{l} \,. \tag{55}$$

$$\Phi(\boldsymbol{\lambda}) = \sum\_{l=0}^{\infty} \Phi(l) \boldsymbol{\lambda}^l,\tag{56}$$

$$\chi(\lambda) = \sum\_{l=0}^{\infty} \chi(l)\lambda^{l}.\tag{57}$$

By applying differential transform on corresponding boundary conditions, we obtained

$$\begin{cases} \begin{aligned} &\stackrel{f}{0}(0)=0, & f(1)=\frac{1}{2}, & g(0)=1, & m(0)=0, & n(0)=0, \\ &\theta(0)=1, & \phi(0)=0, & \chi(0)=0, & \underline{f}(2)=\Pi\_{1}, & f(3)=\Pi\_{2}, \\ &g(1)=\Pi\_{3}, & m(1)=\Pi\_{4}, & n(1)=\Pi\_{5}, & \theta(1)=\Pi\_{6}, & \phi(1)=\Pi\_{6}, \\ &\chi(1)=\Pi\_{8} \end{aligned} \end{cases} \end{cases} \qquad \begin{aligned} \eta(0)=0, \qquad \eta(0)=0, \qquad \chi(0)=0, \\ \chi(0)=0, & \underline{f}(1)=\Pi\_{2}, & \phi(1)=\Pi\_{6}, \\ \underline{\chi}(1)=\underline{\chi}(1), & \underline{\chi}(1)=\underline{\chi}(1), & \underline{\chi}(1)=\underline{\chi}(1), \\ &\underline{\chi}(1)=\underline{\chi}(1), & \underline{\chi}(1)=\underline{\chi}(1), \\ \underline{\chi}(1)=\underline{\chi}(1), & \underline{\chi}(1)=\underline{\chi}(1), & \underline{\chi}(1)=\underline{\chi}(1), \end{aligned}$$

where Π*e* (*e* = 1, . . . , 8) are the constants. Substituting transformations given in Equations (43)–(50) into Equations (30)–(36), and solved with support of associated boundary conditions shown in Equation (58), the resulting solutions in the form of the series are:

$$f(\lambda) = \dot{f}\_1 \lambda^2 + \dot{f}\_2 \lambda^3 + \dot{f}\_3 \lambda^4 + \dot{f}\_4 \lambda^5 + \dots,\tag{59}$$

$$\log(\lambda) = 1 - \dot{\mathcal{g}}\_1 \lambda + \dot{\mathcal{g}}\_2 \lambda^2 + \dot{\mathcal{g}}\_3 \lambda^3 + \dot{\mathcal{g}}\_4 \lambda^4 + \dots \tag{60}$$

$$m(\lambda) = \dot{m}\_1 \lambda + \dot{m}\_2 \lambda^3 + \dot{m}\_3 \lambda^4 + \dot{m}\_4 \lambda^5 + \dots \tag{61}$$

$$n(\lambda) = \dot{n}\_1 \lambda + \dot{n}\_2 \lambda^3 + \dot{n}\_3 \lambda^4 + \dot{n}\_4 \lambda^5 + \dots,\tag{62}$$

$$\overset{\leftarrow}{\theta}(\lambda) = 1 + \dot{\theta}\_1 \lambda + \dot{\theta}\_2 \lambda^2 + \dot{\theta}\_3 \lambda^3 + \dot{\theta}\_4 \lambda^4 + \dots \tag{63}$$

$$\phi(\lambda) = 1 + \dot{\phi}\_1 \lambda + \dot{\phi}\_2 \lambda^2 + \dot{\phi}\_3 \lambda^3 + \dot{\phi}\_4 \lambda^4 + \dots,\tag{64}$$

$$\chi(\lambda) = 1 + \dot{\chi}\_1 \lambda + \dot{\chi}\_2 \lambda^2 + \dot{\chi}\_3 \lambda^3 + \dot{\chi}\_4 \lambda^4 + \dots,\tag{65}$$

where *f i*, .*gi*, .*mi*, .*ni*, *θi*, . *φi* . *χi*; where *i* = (1, 2, 3, ...) are constants. It is not easy to express them here because of their complex and long numerical values. With the assistance of Mathematica computational software, the equation as mentioned above is solved with 30 iterations. However, it failed to obtain a reasonable rate of convergence. The convergence rate of certain sequences can be improved with certain techniques. Many researchers used the Padé technique, which was used in the form of a rational fraction, i.e., ratio of two polynomials. The results obtained by DTM, owing to the non-linearity on the governing equations, do not satisfy the boundary conditions at infinity without applying the Padé approximation. The obtained solution by DTM must then be merged with Padéapproximation, which gives a substantial rate of convergence at infinity. According to one's desired exactness, a higher order of approximation is required. Here, [5 × 5] order approximation is applied to Equations (59)–(65), the Padé approximants are as follows.

$$f(\lambda) = \frac{1.744240\lambda^2 - 6.384709\lambda^3 + 7.800949\lambda^4 - 2.873131\lambda^5 + \dots}{\text{\huge{0.0577711\dots}} \text{ - } \pi \text{ \huge{0.0577712\dots}} \text{ - } \alpha \text{ \harrow } \pi \text{ \harrow } \lambda \text{ -- } \alpha \text{ \harrow } \pi \text{ \harrow } \lambda},\tag{66}$$

$$\begin{pmatrix} \ddots \\ \end{pmatrix} \quad \begin{aligned} &1 - 2.775474 \lambda + 1.798969 \lambda^2 + 0.612641 \lambda^3 - 0.047549 \lambda^4 - 0.001808 \lambda^5 + \dots \\ &1 - 0.461931 \lambda - 0.480814 \lambda^2 - 0.030027 \lambda^3 - 0.036507 \lambda^4 - 0.008722 \lambda^5 + \dots \\ & \dots \end{aligned} \quad \begin{aligned} &\dots \\ &1 - 0.461931 \lambda - 0.480814 \lambda^2 - 0.036507 \lambda^4 - 0.008722 \lambda^5 + \dots \\ &\dots \end{aligned} \quad \begin{aligned} &\dots \\ &\dots \end{aligned}$$

$$\mathbf{g}(\lambda) = \frac{\mathbf{x}\_1 \mathbf{x}\_2 \mathbf{x}\_3 \mathbf{x}\_4 \mathbf{x}\_5 \mathbf{x}\_6 \mathbf{x}\_7 \mathbf{x}\_8 \mathbf{x}\_9 \mathbf{x}\_2 \mathbf{x}\_4 \mathbf{x}\_3 \mathbf{x}\_4 \mathbf{x}\_5 \mathbf{x}\_4}{1 + 0.580516\lambda + 0.110708\lambda^2 + 0.014909\lambda^3 + 0.003980\lambda^4 + 0.003613\lambda^5 + \dots} \,\tag{67}$$
 
$$\dots \quad 0.706586\lambda - 0.052291\lambda^2 - 0.0453936\lambda^3 + 0.272736\lambda^4 - 0.322963\lambda^5 + \dots} \,\tag{68}$$

$$m(\lambda) = \frac{0.706586\lambda - 0.052291\lambda^2 - 0.0453936\lambda^3 + 0.272736\lambda^4 - 0.322963\lambda^5 + \dots}{1 - 0.074006\lambda - 0.397576\lambda^2 + 0.119954\lambda^3 - 0.060980\lambda^4 - 0.027780\lambda^5 + \dots} \tag{68}$$

$$n(\lambda) = \frac{0.767837\lambda + 1.046017\lambda^2 + 0.365143\lambda^3 + 0.429179\lambda^4 + 0.171075\lambda^5 + \dots}{1 + 1.362290\lambda + 0.039499\lambda^2 + 0.254792\lambda^3 + 0.340033\lambda^4 - 0.212403\lambda^5 + \dots},\tag{69}$$

$$\widetilde{\theta}(\lambda) = \frac{1 - 0.794545\lambda - 0.240481\lambda^2 + 0.053229\lambda^3 - 0.032680\lambda^4 + 0.014409\lambda^5 + \dots}{1.0 + 0.038878\lambda - 0.035237\lambda^2 + 0.050992\lambda^3 - 0.033598\lambda^4 + 0.00129\lambda^5 + \dots} \tag{70}$$

$$\phi(\lambda) = \frac{1 - 1.715367\lambda + 0.560355\lambda^2 + 0.391456\lambda^3 - 0.354482\lambda^4 + 0.119370\lambda^5 + \dots}{1 + 0.217143\lambda - 0.068712\lambda^2 + 0.047085\lambda^3 - 0.043507\lambda^4 - 0.008306\lambda^5 + \dots},\tag{71}$$

$$\chi(\lambda) = \frac{1 - 0.776897\lambda + 0.662042\lambda^2 - 0.785269\lambda^3 + 0.099751\lambda^4 - 0.193734\lambda^5 + \dots}{1 + 2.462760401 + 2.462037\lambda^2 + 1.100651\lambda^3 + 0.1247601\lambda^4 - 0.02767271\lambda^5 + \dots},\tag{72}$$

$$
\lambda^{(\prime \prime \prime)} = 1 + 2.4876949 \lambda + 2.462925 \lambda^2 + 1.100656 \lambda^3 + 0.134289 \lambda^4 - 0.0376572 \lambda^5 + \dots \\ \tag{7.7}
$$
